Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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Interpreting the table of classification of the partitions of $n$

I am going through A NON-RECURSIVE EXPRESSION FOR THE NUMBER OF IRREDUCIBLE REPRESENTATIONS OF THE SYMMETRIC GROUP $S_n$ by AMUNATEGUI. In table I, the classification of the partitions of n according ...
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4answers
135 views

Group conjecture

Conjecture: Given a finite group $G$ and a subset $A\subset G$. Then $\{A,A^2,A^3,\dots\}$ is a group iff $\forall n\in \mathbb N: |A^n|=|A^{n+1}|$. Given that the composition between the ...
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2answers
52 views

Sylow subgroups of $\text{SL}_2(q)$.

Let $p,q$ be primes such that $p$ is a divisor of $|\text{SL}_2(q)|=(q-1)q(q+1)$. Hence $\text{SL}_2(q)$ admits non-trivial Sylow subgroups. I am interested in the isomorphism type of the $p$-Sylow. ...
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1answer
41 views

If $M$ is normal and $M \cap U = U'$ for some special subgroup $U$, then $M / G'$ is a Hall-subgroup of $G / G'$.

Let $G$ be a finite group and $U \le G$ be a finite group of odd order. Suppose that $N_G(U) = TU$ where $T = \langle t \rangle$ for some involution $t \notin U$. Also suppose $U^g \ne U$ implies $U^g ...
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12 views

Presentation for Special $p$-groups

Is a uniform presentation for special $p$ groups of rank 2 known? Thanks in advance.
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0answers
51 views

A rather special monoid

While implementing an embryo of computational algebra on my blog I ran into a rather special monoid and I wonder if it's studied before. After implementing a very simple concept of dynamic sets I ...
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2answers
48 views

Minimal Galois extension, describe structure of $Gal(L/\mathbb Q)$

Find the minimal Galois extension $L$ of $\mathbb Q$ containing $\mathbb Q(\sqrt[4]{5})$. Describe the structure of $Gal(L/\mathbb Q)$. I think $L$ is a splitting field of $X^4-5$ over $\mathbb Q$. ...
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219 views

An inequality for the minimal number of generators of a finite group

Let $G$ be a finite group, $n(G)$ the minimal number of generators and $m(G)$ the minimal number of irreducible complex representations generating (with $\otimes$ and $\oplus$) the left regular ...
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2answers
31 views

Proving associative property, floor function

I need to prove the following operation is associative: $x*y = xy \pmod 5$ I came up with the equation that $x*y=xy-5[\![xy/5]\!]$ I'm having difficulty proving that $x*(yz)=(xy)*z$. After ...
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1answer
45 views

Is it known whether all prime powers $p^k$ with $k\ge 8$ are group-abundant?

Denote the number of groups of order $n$ by $gnu(n)$. A natural number $n\ge 1$ is called group-abundant, if $gnu(n)>n$, group-perfect, if $gnu(n)=n$ and group-deficient, if $gnu(n)<n$. I ...
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1answer
27 views

Subgroups order $p$ in a non-cyclic abelian finite p-group.

Is it true that if $G$ is a finite abelian non-cyclic $p$-group then a subgroup of order $p$ cannot be unique? How can I prove it if the sentence is correct? Excuse me for the question, but I've some ...
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31 views

A character on a subgroup could be written as a difference with some character on the whole group, help on argumentation

Let $G$ be a finite group, and $U \le G$ of odd order such that $N_G(U) = TU$ with $T = \langle t \rangle$ for some involution $t$ and assume that $U^g \ne U$ implies $U^g \cap U = 1$. Then I can not ...
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1answer
55 views

Finite group with three proper subgroups

The Klein-$4$ group is a finite group with exactly three subgroups $H$ such that $1<H<G$. Conversely, if $G$ is a finite group with exactly three subgroups $H$ such that $1<H<G$, then ...
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2answers
20 views

Factorization and Quadratic Non-Residue

Suppose that I can always factor any number modulo $p$ into factors that are smaller than $f(p)$ where $f$ is some function. Does that imply that the least quadratic non-residue is smaller than ...
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36 views

If every Sylow's subgroup is cyclic then $G$ is supersolvable.

I've this exercise to resolve : prove that if $G$ is a finite group and all its Sylow subgroups are cyclic then G is supersoluble. My solution follows: is it correct? Thanks to everyone for the help! ...
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1answer
31 views

On the decomposition of the group ring $\mathbb Q[G]$ over the rationals if $G$ is finite and cyclic

Let $G$ be a cyclic finite group of order $n$. I tried to determine the structure of the group ring $\mathbb Q[G]$ over the rationals $\mathbb Q$, what I got for even $n$ is $$ \mathbb Q[G] = A ...
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1answer
53 views

At most half of elements in $S_n$ has a square root

Let $S_n$ be the permutation group on $\{1,...,n\}$. Prove that at most half of the elements $g \in S_n$ have a square root, ie an element $h \in S_n$ such that $g = h^2$. This is one of the ...
3
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0answers
34 views

Show that over finite fields of characteristic two the finite group $C_3$ has no nontrivial representation

Let $G = \mathbb Z / 3\mathbb Z$ be the cyclic group of order three. I conjecture that every irreducible representation over a finite field of characteristic $2$ must be trivial. But how to prove ...
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0answers
30 views

Easy to generate subgroups of the symmetric group $S_n$

Which subgroups of the symmetric group $S_n$ can be generated in polynomial or subexponential time?
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40 views

How can we analyse a function defined on a group?

$S_{n}$ is a symmetric group, $X\in S_{n}$, $f(X): S_{n} \rightarrow \mathbb{R}$ E.g. $f(X) = ax_{1}^2+bx_{2}x_{3}+cx_{3}^2$, $X=(x_{1},x_{2},x_{3})^{T}$, $x_{i}\in \{\alpha,\beta, \gamma \}, ...
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0answers
37 views

Finite Simple Groups other than $A_n$ and $\rm{PSL}_n$

The finite simple groups taught in undergraduate or graduate courses are only up to $A_n$ or $\rm{PSL}_n$. Even many undergraduate and graduate texts do not consider simple groups beyond these two ...
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1answer
45 views

Do these two groups contain a $S_3$ as a subgroup?

I have the following groups $$ G_1 = \langle S,T : S^3 = T^2 = (ST)^2 \rangle $$ $$ G_2 = \langle S,T : S^4 = T^6 = (ST)^2 = (S^{-1}T)^2 = 1\rangle, $$ which have order $12$ and $24$ respectively. ...
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1answer
43 views

If $G$ is a direct product of simple groups, then is every simple and normal subgroup isomorphic to some factor

A direct factor $A \le G$ is a subgroup for which there exists some $B \le G$ such that $G = A \times B$. If $G = N_1 \times \ldots \times N_k$ and $K \le G$ is a simple subgroup and a direct factor, ...
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31 views

The size of the automorphism group of a graph

I am going through QUANTUM MECHANICAL ALGORITHMS FOR THE NONABELIAN HIDDEN SUBGROUP PROBLEM by Grigni et a. It is said on page 14 that the size of the automorphism group of a graph is either $1$ or ...
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1answer
28 views

$H,K$ normal subgroups of a finite group $G$ , $G \cong H \times K$ , every element of $H$ commutes with every element of $K$ , then is $G=HK$?

Let $H,K$ be normal subgroups of a finite group $G$ such that $G$ is isomorphic with $H \times K$ and every element of $H$ commutes with every element of $K$ , then is it necessary that $G=HK$ ? ( ...
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0answers
20 views

If $|\mbox{Hom}(G_{ab}, Z(G))| = 1$ and $G$ has no abelian direct factor, then $Z(G) \le G'$.

Let $G$ be a finite group which has no nontrivial abelian factor, i.e. an abelian normal subgroup $A$ such that there exists some $B \le G$ with $G = A \times B$. Also suppose that the only ...
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61 views

Alternative proof: G group of order $p^2$, p prime $\Rightarrow$ $H$ is normal in $G$.

Let $G$ be a group of order $p^2$ where $p$ is prime. If $H$ is a subgroup of order $p$, show that $H$ is normal in $G$. I would like to prove this with the tools that the book has provided up to ...
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1answer
20 views

Semidirect product of two groups defined in terms of a homomorphism

I am going through From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups by Bacon et al. I am having trouble to understand the ...
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1answer
145 views

Is there a typo in the formula or does my GAP-package sglppow fail?

This site deals with group formulas for prime powers $p^k$ for $k\le 7$. The formula for $k=7$ seems to be wrong. I compared the results with GAP and the formula is off by $2453$ for $p=13,17,19,23$ ...
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2answers
155 views

Prove that middle cancellation implies that the group is abelian

Suppose that $G$ is a group with the property that for every choice of elements in $G$, $axb=cxd$ implies $ab=cd$. Prove that $G$ is Abelian. (Middle cancellation implies commutativity). ...
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3answers
152 views

Number of groups of order $9261$?

I checked the odd numbers upto $10\ 000$ , whether they are group-perfect ($gnu(n)=n$ , where $gnu(n)$ is the number of groups of order $n$), and the only case I could not decide is $$9261=3^3\times ...
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2answers
33 views

Showing that elements in a finite group is odd and even

Let G be a finite group. Show that the number of elements x of G such that $x^{3}=e$ is odd. Show that the number of elements x of G such that $x^{2}\neq$ e is even Looking for a ...
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29 views

How to prove $M\otimes_{\,G}A≅ {Hom}_{G}({Hom}_{\,\mathbb Z}(M,{\,\mathbb Z}),A)$?

Given that G is a finite group, M is a finitely generated right free G-module and A is a left G-module, there exists a natural G-isomorphism $\phi\ : M\otimes_{\,G}A\rightarrow ...
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1answer
57 views

Every group of order $60$ , having a normal subgroup of order $2$ , has a normal subgroup of order $6$ (without Sylow )?

How to prove , without using Sylow's theorems , that every group of order $60$ , having a normal subgroup of order $2$ , contains a normal subgroup of order $6$ ? Please help . Thanks in advance
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21 views

How to prove the set of all permutations of a finite number of objects forms a group - differently?

Consider the set of all permutation vectors of n distinct numbers, say 1,2,3, ....n. Define + operation on this set c= a+b for every a and b from the set as c(i) = a(b(i)). I have checked with ...
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2answers
55 views

Showing $\left ( a^{-1} ba\right )^{n}=a^{-1}b^{n}a$

Prove that $\left ( a^{-1} ba\right )^{n}=a^{-1}b^{n}a$. Here's what I have: for positive $n$, $\left ( a^{-1}ba \right )^{n}=a^{n}b^{n}a^{-n}$ by the socks and shoes property. Any hints for me ...
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2answers
23 views

How many of these cosets of $\mathbb{Z}^2/H$ are distinct?

My answer: Since all of $(1\; 6), (3\; 5)$ and $(7\; 11)$ are found in $H$, then none of them are distinct. Does this make sense? I'm not sure I understood the question.
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2answers
24 views

Find the number of elements in each factorgroup.

(a) and (c) are 2 and 3 and I don't think I have a problem with those. However for (b) I get 11 cosets but they are not disjoint. According to theory there should only be 6. So what is happening ...
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1answer
50 views

Non-isomorphic groups with identical structure-description

I constructed the non-abelian groups of order $16$ and listed the structure descriptions. The result was : ...
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26 views

Number of Elements of Some Primer Order in a Group [duplicate]

Given a finite group $ G $ of order $ n $, let $ a \in G, |a| = p$, where $p$ is a prime. Show that the number of elements in $G$ of order $p$ is $p - 1$ Hint? Thanks.
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1answer
31 views

Find all permutations such that $\sigma=a\tau a^{-1}$

For (b) and (c) we note that $\sigma$ and $\tau$ have different parity so there cannot be any $a\in S_4$ that will fix that parity mismatch. For (a) we have the cycle $a^{-1}=(3 2 4)$ and it is ...
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3answers
30 views

Is this group index infinite?

Let $G=\left(\begin{matrix}a&b\\c&d\end{matrix}\right)$ where $a,b,c,d\in\mathbb{Z}$ and determinant = 1, be a group. And let $H=\left(\begin{matrix}1&n\\0&1\end{matrix}\right)$ where ...
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0answers
17 views

If $H \le G$ commutes with every subgroup of some family, then it commutes with the subgroup generated by that family. Proof Verification.

Let $\mathcal X = \{ X_i : i \in X \}$ be a family of subgroups for some arbitrary index set $I$. Then $\langle \mathcal X\rangle$ denotes the subgroup generated by all $X_i, i \in I$, i.e. the ...
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1answer
45 views

All elements of this finite abelian group.

$$A=\left(\begin{matrix}1 & 2 & 2 \\ 2 &2&2\\3&4&2 \end{matrix}\right)$$ Let $H$ be a subgroup of $\mathbb{Z}^3$ generated by the vectors $\vec{g_i} = \sum_{j=1}^3 ...
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31 views

$f:G\rightarrow G$ an isomorphism.If $f$ has no nontrivial fixed points and if $f\circ f$ is identity function,then$f(x)=x^{-1}$. [duplicate]

Let $f:G\rightarrow G$ be an isomorphism from a finite group $G$ to itself. If $f$ has no nontrivial fixed points and if $f\circ f$ is the identity function, then $f(x)=x^{-1}$ for all $x\in G$. ...
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3answers
22 views

Proving that this group is Abelian via inverse property

Prove that a group G is Abelian If and Only If $\left ( ab \right )^{-1}=a^{-1}b^{-1}$ for all a and b in G Proving first the If condition: $\left ( ab \right )\left ( ab \right )^{-1}=\left ( ab ...
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25 views

Conjugacy class sizes and classification of finite simple groups

Given a finite group $G$, let $1,n_1,n_2,\cdots, n_k$ denote all the possible sizes of conjugacy classes of $G$, with $1<n_1<n_2\cdots$. The first remarkable theorem by concerning such sequence ...
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1answer
37 views

A direct proof that for finite $G$ we have “$G$ abelian iff all irreducible characters are linear”

A finite group is abelian iff all its irreducible characters have dimension one (hence are linear). A common proof uses that the number of irreducible representations equals the number of conjugacy ...
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29 views

Example of group that doesn't admit a p-complement.

I've this example. Let $H$ a finite, simple, non abelian group, $P$ a non abelian finite p-group and consider the direct product $G=H\times$$P$. Then $N_G(P)/C_G(P)$ is a p-group but $G$ is not ...
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2answers
33 views

How many elements in this group - Q8?

How many elements are there in $Q_8=[x,y ; x^4=1, x^2=y^2, xy=yx^3]$ Am I right in thinking this contains the elements: $x,x^2,x^3,y,y^2,y^3,yx,y^2x,y^3x,yx^2,yx^3,y^2x^2,y^2x^3,y^3x^2$ ?